"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Forms and Galois cohomology

Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence

where is a Galois extension, is the Galois group of (thinking of as an object of the category of field extensions of at all times), is a category of “objects over ,” and is a category of “objects over .”

In fact this description is probably only correct if is a finite Galois extension; if is infinite it should probably be modified by requiring that every function of that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on . To avoid this difficulty we’ll stick to the case that is a finite extension.

Today we’ll recover from this abstract description the somewhat more concrete punchline that -forms of an object can be classified by Galois cohomology , and we’ll give some examples.

Basepoints

Way back when we defined a homotopy fixed point structure, we saw that it could be regarded as a generalization of a 1-cocycle, which it reduces to in the special case that the action of on a category can be strictified so that it becomes an action of on the automorphism group of a single object . This is possible in concrete examples; for example, if is the standard -dimensional vector space over , then , and the action of the Galois group on this is just componentwise. When is it possible abstractly?

Topologically, we’re starting from an action of a group on a space (in our examples, a groupoid), and we want to know when such an action gives rise to an action on the fundamental group . Naively, this is only possible if is fixed by the action of . But this is not a homotopy-theoretic condition, and the homotopy-theoretic refinement is that should be a homotopy fixed point of ; then the action of on the unpointed space can be upgraded to an action of on a pointed space , and taking the fundamental group is a functor on pointed spaces. In fact this defines an equivalence from pointed connected groupoids to groups, and so an equivalence from actions of on pointed connected groupoids to actions of on groups.

For our purposes, what this means is that in order to strictify the Galois action on to an action on for a particular object , we need to pick a homotopy fixed point structure on to act as a basepoint; equivalently, we need to pick a -form . The corresponding homotopy fixed point structure is encoded by maps

as usual, and we can now use these maps to coherently identify each with . The corresponding strictified action of on is given by sending an automorphism to the automorphism

which we’ll write as for simplicity.

Fixing this homotopy fixed point structure allows us to describe other homotopy fixed point structures by describing their difference, which we’ll write as

.

After some simplification, we compute that satisfies the compatibility condition that

(where as usual we’ll need to interpret compositions in diagrammatic order for consistency), which is the usual definition of a 1-cocycle on with coefficients in (with respect to the action defined above). Similarly we get that isomorphisms of homotopy fixed point data correspond to 1-cocycles being cohomologous. Hence:

Theorem: Suppose that has at least one -form . Using this -form as a basepoint, isomorphism classes of -forms on can be identified with elements of the Galois cohomology set with respect to the strictified action above.

Note that this set is naturally pointed by the trivial 1-cocycle, whereas the set of isomorphism classes of homotopy fixed points does not have a natural “trivial” object in it. This reinforces the need to pick a basepoint / -form.

Note also that we haven’t yet provided a prescription for actually writing down a -form given homotopy fixed point data on some .

The real and complex numbers

Galois cohomology becomes particularly easy to describe in the special case that (or more generally any quadratic extension), which is already useful for many applications. Here is generated by a single nontrivial element , namely complex conjugation. The action of on will generally also have the interpretation of complex conjugation (e.g. on matrices), and so the data of a -cocycle amounts to (after trivializing ) the data of a single element such that

.

In other words, is an automorphism of whose inverse is its complex conjugate. Two such automorphisms are cohomologous as 1-cocycles iff there is some such that

.

Some examples

In all of the examples below we are claiming without proof that some Galois prestack is in fact a Galois stack.

Example. Let be the Galois stack of vector spaces, and let , with distinguished -form . The strictified Galois action on is componentwise, and this will tell us what the Galois action is in many other examples involving vector spaces with extra structure. Since every -form of must be an -dimensional vector space over and hence must be isomorphic to , we conclude that

When we learn the following: a 1-cocycle is a matrix such that , and the fact that every such 1-cocycle is cohomologous to zero means that every such matrix can be written in the form for some . This recently came up on MathOverflow.

When , and we learn the following: a 1-cocycle is an element such that

(that is, an element of norm ), and the fact that every such 1-cocycle is cohomologous to zero means that every such element can be written in the form

for some . This gives a parameterization

of the set of rational solutions to the Diophantine equation , and in particular when we recover the usual parameterization of Pythagorean triples. (I learned this from Noam Elkies.)

Example. Let be the Galois stack of commutative algebras, and consider . Its automorphism group as an -algebra is with trivial Galois action, so -forms of are classified by

.

Because the Galois action is trivial, this is the set of conjugacy classes of homomorphisms , or equivalently isomorphism classes of actions of on -element sets. Such an isomorphism class is a disjoint union of transitive actions of on -element sets, which by the Galois correspondence can be identified with finite separable extensions of of degree , and in fact it turns out that -forms of are precisely -algebras of the form

where each is a subextension of and . So in this case we more or less get ordinary Galois theory back.

Example. Let be the Galois stack of algebras, not necessarily commutative, and consider . Its automorphism group as an -algebra is with Galois action inherited from , so -forms of are classified by

.

This classification is related to the Brauer group of , namely the part involving those central simple -algebras which become isomorphic to after extension by scalars to . It is also related to Severi-Brauer varieties, which are -forms of projective space.

To connect this to a previous computation, the short exact sequence gives rise to a longish exact sequence part of which goes

.

Since vanishes by Hilbert’s theorem 90, by exactness we conclude that if the relative Brauer group vanishes, then so does ; equivalently, in this case the only -form of is .

Good properties

One last comment. Galois descent gives us a reason to single out certain properties P that some objects satisfying Galois descent (such as algebras or commutative algebras) can have as particularly good: namely, those properties which also satisfy Galois descent. This means that

The extension of scalars of a P-object is P.

The -forms of any P-object are P.

For example, for algebras, being semisimple is not a good property in this sense: the extension of scalars of a semisimple -algebra can fail to be semisimple (e.g. if is an extension of which is not separable). The good version of this property is being separable, which is equivalent to being “geometrically semisimple” in the sense that is semisimple for all field extensions .

Similarly, for commutative algebras, being isomorphic to a finite product of copies of the ground field is not a good property in this sense: although it is preserved under extension of scalars, since , it is not preserved under taking -forms, as we saw above.

Your guess about infinite Galois extensions is correct. For instance, consider the stack of modules (or quasicoherent sheaves, if you like): it is an fpqc-stack, so it certainly satisfies the (effective) Galois descent condition. But if you unwind what fpqc descent means for a field extension (which is always fpqc!) you will see that it only reduces to homotopy fixed points for a group action in the case of a finite Galois extension.

In a bit more detail: let be a field extension and consider the simplicial kernel of . By fpqc descent, it has an effective quotient (in the category of fpqc sheaves), namely the representable sheaf on , and moreover, sends this colimit diagram to a bilimit diagram. Since we are dealing with objects in a 2-category, we can truncate these (co)simplicial diagrams above degree 2. When is a finite Galois extension, the objects in the simplicial kernel are just disjoint unions of copies of . It is instructive to work out what the face operators are in terms of this identification: at the bottom, one is the diagonal embedding , and the other is , where is some enumeration of the Galois group.

For anyone reading Zhen’s comment, and not particularly enamored by the words ‘simplicial kernel’, let me just state another way of seeing it (which I assume both Zhen and Qiaochu know well). Namely, fpqc descent along involves, in particular, . Now, is fpqc for any a field extension. What makes the finite case different is that, there, $\displaystyle L\otimes_k L$ is just with the map being .

Thus, fpqc descent works for any extension, but explicitly being able to relate it to the Galois group is a failed endeavor for non-finite Galois extensions. This also suggestions how one might replace this notion by studying stacks on the subcategory of consisting of Galois objects, for other .

Again, so Zhen doesn’t yell at me, this is the exact same thing he wrote (I assume) in different language.🙂